Matrices

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which is a (“two by two”) matrix. A general matrix may be
,
where is the
number of rows, and
is the
number of columns. For
example, the general matrix is

(Either square brackets or large parentheses may be used.)
Theth
element7.3 of a matrix may be
denoted by
or .
The rows and columns of matrices are
normally numbered from instead of from; thus,
and .
When , the
matrix is said to be square.

The transpose of a real
matrix is denoted by and is defined by

Note that while
is ,
its transpose is .

A complex matrix, is simply a matrix containing
complex numbers. The transpose of a complex matrix is normally defined to
includeconjugation. The conjugating transpose operation is
called theHermitian transpose. To avoid confusion, in this tutorial, and the word “transpose” will always denote
transpositionwithout conjugation, while conjugating
transposition will be denoted by and be called the “Hermitian transpose” or the “conjugate
transpose.” Thus,

(as we have been using) is a
matrix. In contexts where matrices are being used (only this
section for this course), it is best to define all vectors as
column vectors and to indicate row vectors using the
transpose notation, as was done in the equation above.